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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
Eighth Edition
Ferdinand P. BeerE. Russell Johnston, Jr.
Lecture Notes:J. Walt OlerTexas Tech University
CHAPTER
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
11Kinematics of Particles
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 2
ContentsIntroductionRectilinear Motion: Position,
Velocity & AccelerationDetermination of the Motion of a
ParticleSample Problem 11.2Sample Problem 11.3Uniform Rectilinear-MotionUniformly Accelerated Rectilinear-
MotionMotion of Several Particles: Relative
MotionSample Problem 11.4Motion of Several Particles:
Dependent Motion
Sample Problem 11.5
Graphical Solution of Rectilinear-Motion Problems
Other Graphical MethodsCurvilinear Motion: Position, Velocity &
AccelerationDerivatives of Vector FunctionsRectangular Components of Velocity and
AccelerationMotion Relative to a Frame in TranslationTangential and Normal ComponentsRadial and Transverse ComponentsSample Problem 11.10Sample Problem 11.12
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 3
Introduction• Dynamics includes:
- Kinematics: study of the geometry of motion. Kinematics is used to relate displacement, velocity, acceleration, and time without reference to the cause of motion.
- Kinetics: study of the relations existing between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by given forces or to determine the forces required to produce a given motion.
• Rectilinear motion: position, velocity, and acceleration of a particle as it moves along a straight line.
• Curvilinear motion: position, velocity, and acceleration of a particle as it moves along a curved line in two or three dimensions.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 4
Rectilinear Motion: Position, Velocity & Acceleration
• Particle moving along a straight line is said to be in rectilinear motion.
• Position coordinate of a particle is defined by positive or negative distance of particle from a fixed origin on the line.
• The motion of a particle is known if the position coordinate for particle is known for every value of time t. Motion of the particle may be expressed in the form of a function, e.g.,
326 ttx −=
or in the form of a graph x vs. t.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 5
Rectilinear Motion: Position, Velocity & Acceleration
• Instantaneous velocity may be positive or negative. Magnitude of velocity is referred to as particle speed.
• Consider particle which occupies position P at time t and P’ at t+Δt,
txv
tx
t ΔΔ
==
ΔΔ
=
→Δ 0lim
Average velocity
Instantaneous velocity
• From the definition of a derivative,
dtdx
txv
t=
ΔΔ
=→Δ 0
lim
e.g.,
2
32
312
6
ttdtdxv
ttx
−==
−=
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 6
Rectilinear Motion: Position, Velocity & Acceleration• Consider particle with velocity v at time t and
v’ at t+Δt,
Instantaneous accelerationtva
t ΔΔ
==→Δ 0
lim
tdtdva
ttvdt
xddtdv
tva
t
612
312e.g.
lim
2
2
2
0
−==
−=
==ΔΔ
=→Δ
• From the definition of a derivative,
• Instantaneous acceleration may be:- positive: increasing positive velocity
or decreasing negative velocity- negative: decreasing positive velocity
or increasing negative velocity.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 7
Rectilinear Motion: Position, Velocity & Acceleration
• Consider particle with motion given by326 ttx −=
2312 ttdtdxv −==
tdt
xddtdva 6122
2−===
• at t = 0, x = 0, v = 0, a = 12 m/s2
• at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0
• at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2
• at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 8
Determination of the Motion of a Particle• Recall, motion of a particle is known if position is known for all time t.
• Typically, conditions of motion are specified by the type of acceleration experienced by the particle. Determination of velocity and position requires two successive integrations.
• Three classes of motion may be defined for:- acceleration given as a function of time, a = f(t)- acceleration given as a function of position, a = f(x)- acceleration given as a function of velocity, a = f(v)
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 9
Determination of the Motion of a Particle• Acceleration given as a function of time, a = f(t):
( ) ( )( )
( ) ( ) ( )
( ) ( )( )
( ) ( ) ( )∫∫∫
∫∫∫
=−===
=−====
tttx
x
tttv
v
dttvxtxdttvdxdttvdxtvdtdx
dttfvtvdttfdvdttfdvtfadtdv
00
0
00
0
0
0
• Acceleration given as a function of position, a = f(x):
( )
( )( )
( ) ( ) ( )∫∫∫ =−==
=====
x
x
x
x
xv
vdxxfvxvdxxfdvvdxxfdvv
xfdxdvva
dtdva
vdxdt
dtdxv
000
202
1221
or or
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 10
Determination of the Motion of a Particle
• Acceleration given as a function of velocity, a = f(v):
( )( ) ( )
( )
( )( )
( )( )
( )
( )( )
( )( )
( )∫
∫∫
∫
∫∫
=−
====
=
====
tv
v
tv
v
tx
x
tv
v
ttv
v
vfdvvxtx
vfdvvdx
vfdvvdxvfa
dxdvv
tvf
dv
dtvf
dvdtvf
dvvfadtdv
0
00
0
0
0
0
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 11
Sample Problem 11.2
Determine:• velocity and elevation above ground at
time t, • highest elevation reached by ball and
corresponding time, and • time when ball will hit the ground and
corresponding velocity.
Ball tossed with 10 m/s vertical velocity from window 20 m above ground.
SOLUTION:
• Integrate twice to find v(t) and y(t).
• Solve for t at which velocity equals zero (time for maximum elevation) and evaluate corresponding altitude.
• Solve for t at which altitude equals zero (time for ground impact) and evaluate corresponding velocity.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 12
Sample Problem 11.2
( )( ) tvtvdtdv
adtdv
ttv
v81.981.9
sm81.9
00
2
0
−=−−=
−==
∫∫
( ) ttv ⎟⎠⎞
⎜⎝⎛−= 2s
m81.9sm10
( )( ) ( ) 2
21
00
81.91081.910
81.910
0
ttytydttdy
tvdtdy
tty
y−=−−=
−==
∫∫
( ) 22s
m905.4sm10m20 ttty ⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=
SOLUTION:• Integrate twice to find v(t) and y(t).
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 13
Sample Problem 11.2• Solve for t at which velocity equals zero and evaluate
corresponding altitude.
( ) 0sm81.9
sm10 2 =⎟
⎠⎞
⎜⎝⎛−= ttv
s019.1=t
• Solve for t at which altitude equals zero and evaluate corresponding velocity.
( )
( ) ( )22
22
s019.1sm905.4s019.1
sm10m20
sm905.4
sm10m20
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=
y
ttty
m1.25=y
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 14
Sample Problem 11.2• Solve for t at which altitude equals zero and
evaluate corresponding velocity.
( ) 0sm905.4
sm10m20 2
2 =⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+= ttty
( )s28.3
smeaningles s243.1=−=
tt
( )
( ) ( )s28.3sm81.9
sm10s28.3
sm81.9
sm10
2
2
⎟⎠⎞
⎜⎝⎛−=
⎟⎠⎞
⎜⎝⎛−=
v
ttv
sm2.22−=v
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 15
Sample Problem 11.3
Brake mechanism used to reduce gun recoil consists of piston attached to barrel moving in fixed cylinder filled with oil. As barrel recoils with initial velocity v0 , piston moves and oil is forced through orifices in piston, causing piston and cylinder to decelerate at rate proportional to their velocity.
Determine v(t), x(t), and v(x).
kva −=
SOLUTION:
• Integrate a = dv/dt = -kv to find v(t).
• Integrate v(t) = dx/dt to find x(t).
• Integrate a = v dv/dx = -kv to find v(x).
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 16
Sample Problem 11.3SOLUTION:
• Integrate a = dv/dt = -kv to find v(t).( ) ( ) kt
vtvdtk
vdvkv
dtdva
ttv
v−=−=−== ∫∫
00ln
0
( ) ktevtv −= 0
• Integrate v(t) = dx/dt to find x(t).
( )
( )( )
tkt
tkt
tx
kt
ek
vtxdtevdx
evdtdxtv
00
00
0
0
1⎥⎦⎤
⎢⎣⎡−==
==
−−
−
∫∫
( ) ( )ktekvtx −−= 10
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 17
Sample Problem 11.3• Integrate a = v dv/dx = -kv to find v(x).
kxvv
dxkdvdxkdvkvdxdvva
xv
v
−=−
−=−=−== ∫∫
0
00
kxvv −= 0
• Alternatively,
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
0 1v
tvkvtx
kxvv −= 0
( ) ( )0
0 or v
tveevtv ktkt == −−
( ) ( )ktekvtx −−= 10with
and
then
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 18
Uniform Rectilinear Motion
For particle in uniform rectilinear motion, the acceleration is zero and the velocity is constant.
vtxxvtxx
dtvdx
vdtdx
tx
x
+==−
=
==
∫∫
0
0
00
constant
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 19
Uniformly Accelerated Rectilinear MotionFor particle in uniformly accelerated rectilinear motion, the acceleration of
the particle is constant.
atvv
atvvdtadvadtdv tv
v
+=
=−=== ∫∫
0
000
constant
( )
221
00
221
000
000
attvxx
attvxxdtatvdxatvdtdx tx
x
++=
+=−+=+= ∫∫
( ) ( )
( )020
2
020
221
2
constant00
xxavv
xxavvdxadvvadxdvv
x
x
v
v
−+=
−=−=== ∫∫
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 20
Motion of Several Particles: Relative Motion• For particles moving along the same line, time
should be recorded from the same starting instant and displacements should be measured from the same origin in the same direction.
=−= ABAB xxx relative position of B with respect to A
ABAB xxx +=
=−= ABAB vvv relative velocity of B with respect to A
ABAB vvv +=
=−= ABAB aaa relative acceleration of B with respect to A
ABAB aaa +=
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 21
Sample Problem 11.4
Ball thrown vertically from 12 m level in elevator shaft with initial velocity of 18 m/s. At same instant, open-platform elevator passes 5 m level moving upward at 2 m/s.
Determine (a) when and where ball hits elevator and (b) relative velocity of ball and elevator at contact.
SOLUTION:
• Substitute initial position and velocity and constant acceleration of ball into general equations for uniformly accelerated rectilinear motion.
• Substitute initial position and constant velocity of elevator into equation for uniform rectilinear motion.
• Write equation for relative position of ball with respect to elevator and solve for zero relative position, i.e., impact.
• Substitute impact time into equation for position of elevator and relative velocity of ball with respect to elevator.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 22
Sample Problem 11.4SOLUTION:• Substitute initial position and velocity and constant
acceleration of ball into general equations for uniformly accelerated rectilinear motion.
22
221
00
20
sm905.4
sm18m12
sm81.9
sm18
ttattvyy
tatvv
B
B
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=++=
⎟⎠⎞
⎜⎝⎛−=+=
• Substitute initial position and constant velocity of elevator into equation for uniform rectilinear motion.
ttvyy
v
EE
E
⎟⎠⎞
⎜⎝⎛+=+=
=
sm2m5
sm2
0
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 23
Sample Problem 11.4• Write equation for relative position of ball with respect to
elevator and solve for zero relative position, i.e., impact.
( ) ( ) 025905.41812 2 =+−−+= ttty EB
( )s65.3
smeaningles s39.0=−=
tt
• Substitute impact time into equations for position of elevator and relative velocity of ball with respect to elevator.
( )65.325+=Eym3.12=Ey
( )( )65.381.916
281.918
−=
−−= tv EB
sm81.19−=EBv
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 24
Motion of Several Particles: Dependent Motion• Position of a particle may depend on position of one
or more other particles.
• Position of block B depends on position of block A. Since rope is of constant length, it follows that sum of lengths of segments must be constant.
=+ BA xx 2 constant (one degree of freedom)
• Positions of three blocks are dependent.
=++ CBA xxx 22 constant (two degrees of freedom)
• For linearly related positions, similar relations hold between velocities and accelerations.
022or022
022or022
=++=++
=++=++
CBACBA
CBACBA
aaadt
dvdt
dvdt
dv
vvvdt
dxdt
dxdt
dx
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 25
Sample Problem 11.5
Pulley D is attached to a collar which is pulled down at 3 in./s. At t = 0, collar A starts moving down from K with constant acceleration and zero initial velocity. Knowing that velocity of collar A is 12 in./s as it passes L, determine the change in elevation, velocity, and acceleration of block B when block A is at L.
SOLUTION:
• Define origin at upper horizontal surface with positive displacement downward.
• Collar A has uniformly accelerated rectilinear motion. Solve for acceleration and time t to reach L.
• Pulley D has uniform rectilinear motion. Calculate change of position at time t.
• Block B motion is dependent on motions of collar A and pulley D. Write motion relationship and solve for change of block B position at time t.
• Differentiate motion relation twice to develop equations for velocity and acceleration of block B.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 26
Sample Problem 11.5SOLUTION:• Define origin at upper horizontal surface with
positive displacement downward.
• Collar A has uniformly accelerated rectilinear motion. Solve for acceleration and time t to reach L.
( ) ( )[ ]
( ) 2
20
20
2
sin.9in.82
sin.12
2
==⎟⎠⎞
⎜⎝⎛
−+=
AA
AAAAA
aa
xxavv
( )
s 333.1sin.9
sin.12 2
0
==
+=
tt
tavv AAA
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 27
Sample Problem 11.5• Pulley D has uniform rectilinear motion. Calculate
change of position at time t.( )
( ) ( ) in. 4s333.1s
in.30
0
=⎟⎠⎞
⎜⎝⎛=−
+=
DD
DDD
xx
tvxx
• Block B motion is dependent on motions of collar A and pulley D. Write motion relationship and solve for change of block B position at time t.
Total length of cable remains constant,( ) ( ) ( )
( )[ ] ( )[ ] ( )[ ]( ) ( ) ( )[ ] 0in.42in.8
02
22
0
000
000
=−++
=−+−+−
++=++
BB
BBDDAA
BDABDA
xx
xxxxxx
xxxxxx
( ) in.160 −=− BB xx
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 28
Sample Problem 11.5• Differentiate motion relation twice to develop
equations for velocity and acceleration of block B.
0s
in.32s
in.12
02
constant2
=+⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
=++
=++
B
BDA
BDA
v
vvv
xxx
sin.18=Bv
0sin.9
02
2 =+⎟⎠⎞
⎜⎝⎛
=++
B
BDA
v
aaa
2sin.9−=Ba
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 29
Graphical Solution of Rectilinear-Motion Problems
• Given the x-t curve, the v-t curve is equal to the x-t curve slope.
• Given the v-t curve, the a-t curve is equal to the v-t curve slope.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 30
Graphical Solution of Rectilinear-Motion Problems
• Given the a-t curve, the change in velocity between t1 and t2 is equal to the area under the a-t curve between t1 and t2 .
• Given the v-t curve, the change in position between t1 and t2 is equal to the area under the v-t curve between t1 and t2 .
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 31
Other Graphical Methods• Moment-area method to determine particle position at
time t directly from the a-t curve:
( )∫ −+=
−=−
1
0
110
01 curve under areav
vdvtttv
tvxx
using dv = a dt ,
( )∫ −+=−1
0
11001
v
vdtatttvxx
( ) =−∫1
0
1
v
vdtatt first moment of area under a-t curve
with respect to t = t1 line.
( )( )Ct
tta-ttvxxcentroidofabscissa
curve under area 11001=
−++=
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 32
Other Graphical Methods
• Method to determine particle acceleration from v-x curve:
===
=
BCAB
dxdvva
θtansubnormal to v-x curve
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 33
Curvilinear Motion: Position, Velocity & Acceleration• Particle moving along a curve other than a straight line
is in curvilinear motion.
• Position vector of a particle at time t is defined by a vector between origin O of a fixed reference frame and the position occupied by particle.
• Consider particle which occupies position P defined by at time t and P’ defined by at t + Δt, rr r ′r
=
=ΔΔ
=
=
=ΔΔ
=
→Δ
→Δ
dtds
tsv
dtrd
trv
t
t
0
0
lim
limrr
r
instantaneous velocity (vector)
instantaneous speed (scalar)
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 34
Curvilinear Motion: Position, Velocity & Acceleration
=
=ΔΔ
=→Δ dt
vdtva
t
rrr
0lim
instantaneous acceleration (vector)
• Consider velocity of particle at time t and velocity at t + Δt,
vr
vr′
• In general, acceleration vector is not tangent to particle path and velocity vector.
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Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 35
Derivatives of Vector Functions( )uPr
• Let be a vector function of scalar variable u,( ) ( )
uuPuuP
uP
duPd
uu Δ−Δ+
=ΔΔ
=→Δ→Δ
rrrr
00limlim
• Derivative of vector sum,( )
duQd
duPd
duQPd
rrrr
+=+
( )duPdfP
dudf
duPfd
rr
r
+=
• Derivative of product of scalar and vector functions,
• Derivative of scalar product and vector product,( )
( )duQdPQ
duPd
duQPd
duQdPQ
duPd
duQPd
rrr
rrr
rrr
rrr
×+×=×
•+•=•
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Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 36
Rectangular Components of Velocity & Acceleration• When position vector of particle P is given by its
rectangular components,
kzjyixrrrrr ++=
• Velocity vector,
kvjviv
kzjyixkdtdzj
dtdyi
dtdxv
zyxrrr
r&
r&
r&
rrrr
++=
++=++=
• Acceleration vector,
kajaia
kzjyixkdt
zdjdt
ydidt
xda
zyxrrr
r&&
r&&
r&&
rrrr
++=
++=++= 2
2
2
2
2
2
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Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 37
Rectangular Components of Velocity & Acceleration• Rectangular components particularly effective
when component accelerations can be integrated independently, e.g., motion of a projectile,
00 ==−==== zagyaxa zyx &&&&&&
with initial conditions,( ) ( ) ( ) 0,,0 000000 ==== zyx vvvzyx
Integrating twice yields
( ) ( )( ) ( ) 0
02
21
00
00
=−==
=−==
zgtyvytvx
vgtvvvv
yx
zyyxx
• Motion in horizontal direction is uniform.
• Motion in vertical direction is uniformly accelerated.
• Motion of projectile could be replaced by two independent rectilinear motions.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 38
Motion Relative to a Frame in Translation• Designate one frame as the fixed frame of reference.
All other frames not rigidly attached to the fixed reference frame are moving frames of reference.
• Position vectors for particles A and B with respect to the fixed frame of reference Oxyz are . and BA rr rr
• Vector joining A and B defines the position of B with respect to the moving frame Ax’y’z’ and
ABrr
ABAB rrr rrr +=
• Differentiating twice,=ABvr velocity of B relative to A.ABAB vvv rrr +=
=ABar acceleration of B relative to A.
ABAB aaa rrr +=
• Absolute motion of B can be obtained by combining motion of A with relative motion of B with respect to moving reference frame attached to A.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 39
Tangential and Normal Components• Velocity vector of particle is tangent to path of
particle. In general, acceleration vector is not. Wish to express acceleration vector in terms of tangential and normal components.
• are tangential unit vectors for the particle path at P and P’. When drawn with respect to the same origin, and
is the angle between them.
tt ee rr ′ and
ttt eee rrr −′=ΔθΔ
( )
( )
θ
θθ
θ
θ
θθ
dede
eee
e
tn
nnt
t
rr
rrr
=
=ΔΔ
=ΔΔ
Δ=Δ
→Δ→Δ 22sinlimlim
2sin2
00
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 40
Tangential and Normal Componentstevv rr =• With the velocity vector expressed as
the particle acceleration may be written as
dtds
dsd
dedve
dtdv
dtedve
dtdv
dtvda tt
θθ
rr
rr
rr +=+==
but
vdtdsdsde
ded
nt === θρθ
rr
After substituting,
ρρ
22 vadtdvaeve
dtdva ntnt ==+= rrr
• Tangential component of acceleration reflects change of speed and normal component reflects change of direction.
• Tangential component may be positive or negative. Normal component always points toward center of path curvature.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 41
Tangential and Normal Components
ρρ
22 vadtdvaeve
dtdva ntnt ==+= rrr
• Relations for tangential and normal acceleration also apply for particle moving along space curve.
• Plane containing tangential and normal unit vectors is called the osculating plane.
ntb eee rrr ×=
• Normal to the osculating plane is found from
binormale
normalprincipal e
b
n
=
=r
r
• Acceleration has no component along binormal.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 42
Radial and Transverse Components• When particle position is given in polar coordinates,
it is convenient to express velocity and acceleration with components parallel and perpendicular to OP.
rr e
dede
ded r
rr
r−==
θθθ
θ
dtde
dtd
ded
dted rr θθ
θ θr
rr==
dtde
dtd
ded
dted
rθθ
θθθ rrr
−==
( )
θ
θ
θ
θ
erer
edtdre
dtdr
dtedre
dtdrer
dtdv
r
rr
rrr&r
&
rrr
rrr
+=
+=+==
• The particle velocity vector is
• Similarly, the particle acceleration vector is
( ) ( ) θ
θθθ
θ
θθθ
θθθ
θ
errerr
dted
dtdre
dtdre
dtd
dtdr
dted
dtdre
dtrd
edtdre
dtdr
dtda
r
rr
r
r&&&&r&&&
rrr
rr
rrr
22
2
2
2
2
++−=
++++=
⎟⎠⎞
⎜⎝⎛ +=
rerr rr =
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 43
Radial and Transverse Components• When particle position is given in cylindrical
coordinates, it is convenient to express the velocity and acceleration vectors using the unit vectors . and ,, keeR
rrrθ
• Position vector,
kzeRr Rrrr +=
• Velocity vector,
kzeReRdtrdv R
r&
r&r&r
r ++== θθ
• Acceleration vector,
( ) ( ) kzeRReRRdtvda R
r&&
r&&&&r&&&r
r +++−== θθθθ 22
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 44
Sample Problem 11.10
A motorist is traveling on curved section of highway at 60 mph. The motorist applies brakes causing a constant deceleration rate.
Knowing that after 8 s the speed has been reduced to 45 mph, determine the acceleration of the automobile immediately after the brakes are applied.
SOLUTION:
• Calculate tangential and normal components of acceleration.
• Determine acceleration magnitude and direction with respect to tangent to curve.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 45
Sample Problem 11.10
ft/s66mph45ft/s88mph60
==
SOLUTION:• Calculate tangential and normal components of
acceleration.( )
( )2
22
2
sft10.3
ft2500sft88
sft75.2
s 8sft8866
===
−=−
=ΔΔ
=
ρva
tva
n
t
• Determine acceleration magnitude and direction with respect to tangent to curve.
( ) 2222 10.375.2 +−=+= nt aaa 2sft14.4=a
75.210.3tantan 11 −− ==
t
naaα °= 4.48α
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 46
Sample Problem 11.12
Rotation of the arm about O is defined by θ = 0.15t2 where θ is in radians and t in seconds. Collar B slides along the arm such that r = 0.9 - 0.12t2 where r is in meters.
After the arm has rotated through 30o, determine (a) the total velocity of the collar, (b) the total acceleration of the collar, and (c) the relative acceleration of the collar with respect to the arm.
SOLUTION:
• Evaluate time t for θ = 30o.
• Evaluate radial and angular positions, and first and second derivatives at time t.
• Calculate velocity and acceleration in cylindrical coordinates.
• Evaluate acceleration with respect to arm.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 47
Sample Problem 11.12SOLUTION:• Evaluate time t for θ = 30o.
s 869.1rad524.0300.15 2
==°==
ttθ
• Evaluate radial and angular positions, and first and second derivatives at time t.
2
2
sm24.0
sm449.024.0m 481.012.09.0
−=
−=−==−=
r
trtr
&&
&
2
2
srad30.0
srad561.030.0rad524.015.0
=
==
==
θ
θθ
&&
& tt
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 48
Sample Problem 11.12• Calculate velocity and acceleration.
( )( )
rr
r
vvvvv
rvsrv
θθ
θ
β
θ
122 tan
sm270.0srad561.0m481.0m449.0
−=+=
===
−==&
&
°== 0.31sm524.0 βv
( )( )
( )( ) ( )( )
rr
r
aaaaa
rra
rra
θθ
θ
γ
θθ
θ
122
2
2
2
22
2
tan
sm359.0
srad561.0sm449.02srad3.0m481.0
2sm391.0
srad561.0m481.0sm240.0
−=+=
−=
−+=
+=
−=
−−=
−=
&&&&
&&&
°== 6.42sm531.0 γa
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
EighthEdition
11 - 49
Sample Problem 11.12• Evaluate acceleration with respect to arm.
Motion of collar with respect to arm is rectilinear and defined by coordinate r.
2sm240.0−== ra OAB &&